DBMS _Relational model

Database System Concepts, 6th
Ed.
©Silberschatz, Korth and Sudarshan
See www.db-book.com for conditions on re-use
Chapter 2: Intro to RelationalChapter 2: Intro to Relational
ModelModel

©Silberschatz, Korth and Sudarshan2.2Database System Concepts - 6th
Edition
Example of a RelationExample of a Relation
attributes
(or columns)
tuples
(or rows)

Edition
Attribute TypesAttribute Types
The set of allowed values for each attribute is called the domain
of the attribute
Attribute values are (normally) required to be atomic; that is,
indivisible
The special value null is a member of every domain. Indicated
that the value is “unknown”
The null value causes complications in the definition of many
operations

Edition
Relation Schema and InstanceRelation Schema and Instance
A1, A2, …, An are attributes
R = (A1, A2, …, An ) is a relation schema
Example:
instructor = (ID, name, dept_name, salary)
Formally, given sets D1, D2, …. Dn a relation r is a subset of
D1 x D2 x … x Dn
Thus, a relation is a set of n-tuples (a1, a2, …, an) where each ai ∈ Di
The current values (relation instance) of a relation are specified
by a table
An element t of r is a tuple, represented by a row in a table

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Relations are UnorderedRelations are Unordered
Order of tuples is irrelevant (tuples may be stored in an arbitrary order)
Example: instructor relation with unordered tuples

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KeysKeys
Let K ⊆ R
K is a superkey of R if values for K are sufficient to identify a unique
tuple of each possible relation r(R)
Example: {ID} and {ID,name} are both superkeys of instructor.
Superkey K is a candidate key if K is minimal
Example: {ID} is a candidate key for Instructor
One of the candidate keys is selected to be the primary key.
which one?
Foreign key constraint: Value in one relation must appear in another
Referencing relation
Referenced relation
Example – dept_name in instructor is a foreign key from instructor
referencing department

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Schema Diagram for University DatabaseSchema Diagram for University Database

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Relational Query LanguagesRelational Query Languages
Procedural vs .non-procedural, or declarative
“Pure” languages:
Relational algebra
Tuple relational calculus
Domain relational calculus
The above 3 pure languages are equivalent in computing power
We will concentrate in this chapter on relational algebra
Not turning-machine equivalent
consists of 6 basic operations

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Select Operation – selection of rowsSelect Operation – selection of rows
(tuples)(tuples)
Relation r
σA=B ^ D > 5 (r)

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Project Operation – selection of columnsProject Operation – selection of columns
(Attributes)(Attributes)
Relation r:
∏A,C (r)

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Union of two relationsUnion of two relations
Relations r, s:
r ∪ s:

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Set difference of two relationsSet difference of two relations
Relations r, s:
r – s:

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Set intersection of two relationsSet intersection of two relations
Relation r, s:
r ∩ s
Note: r ∩ s = r – (r – s)

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joining two relations -- Cartesian-joining two relations -- Cartesian-
productproduct
Relations r, s:
r x s:

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Cartesian-product – naming issueCartesian-product – naming issue
Relations r, s:
r x s: s.B
B
r.B

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Renaming a TableRenaming a Table
Allows us to refer to a relation, (say E) by more than one name.
ρ x (E)
returns the expression E under the name X
Relations r
r x ρ s (r)
α
α
β
β
1
1
2
2
α
β
α
β
1
2
1
2
r.A r.B s.A s.B

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Composition of OperationsComposition of Operations
Can build expressions using multiple operations
Example: σA=C (r x s)
r x s
σA=C (r x s)

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Joining two relations – Natural JoinJoining two relations – Natural Join
Let r and s be relations on schemas R and S respectively.
Then, the “natural join” of relations R and S is a relation on
schema R ∪ S obtained as follows:
Consider each pair of tuples tr from r and ts from s.
If tr and ts have the same value on each of the attributes in
R ∩ S, add a tuple t to the result, where
 t has the same value as tr on r
 t has the same value as ts on s

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Natural Join ExampleNatural Join Example
Relations r, s:
Natural Join
r s
∏ A, r.B, C, r.D, E (σ r.B = s.B r.D = s.D˄ (r x s)))

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Notes about Relational LanguagesNotes about Relational Languages
Each Query input is a table (or set of tables)
Each query output is a table.
All data in the output table appears in one of the input tables
Relational Algebra is not Turning complete
Can we compute:
SUM
AVG
MAX
MIN

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Summary of Relational Algebra OperatorsSummary of Relational Algebra Operators
Symbol (Name) Example of Use
(Selection) σ
salary > = 85000 (instructor)
σ
Return rows of the input relation that satisfy the predicate.
Π
(Projection) Π
ID, salary (instructor)
Output specified attributes from all rows of the input relation. Remove
duplicate tuples from the output.
x
(Cartesian Product) instructor x department
Output pairs of rows from the two input relations that have the same value on
all attributes that have the same name.
∪
(Union) Π
name (instructor) ∪ Π
name (student)
Output the union of tuples from the two input relations.
(Natural Join) instructor ⋈ department
Output pairs of rows from the two input relations that have the same value on
all attributes that have the same name.
⋈
-
(Set Difference) Π
name (instructor) -- Π
name (student)
Output the set difference of tuples from the two input relations.

Database System Concepts, 6th
Ed.
©Silberschatz, Korth and Sudarshan
See www.db-book.com for conditions on re-use
End of Chapter 2End of Chapter 2

DBMS _Relational model

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